KPSVD Visualization: Van Loan-Pitsianis Method

Parameters

Original Image: 400 × 400
KPSVD Dimensions: p=40, q=40, r=10, s=10
Rearranged Matrix R(M): 1600 × 100
Available k values: 10, 20, 30

Original Audio

Original Audio Matrix
Original Audio (matrix visualization)
Shape: (400, 400)
Total elements: 160000

k=10 Approximation

Approximation k=10
k=10 Approximation
Compression Ratio: 9.41×
Storage: 17010 elements (vs 160000 original)
MSE: 41.89
PSNR: 31.91 dB

Left Factor Noise Series (k=10)

Noise added to U (left factor), noise levels: [0.0, 0.01, 0.03]

Left noise 0
U noise σ=0.0
PSNR: 31.91 dB
Left noise 1
U noise σ=0.01
PSNR: 12.87 dB
Left noise 2
U noise σ=0.03
PSNR: 3.49 dB

Right Factor Noise Series (k=10)

Noise added to V (right factor), noise levels: [0.0, 0.01, 0.03]

Right noise 0
V noise σ=0.0
PSNR: 31.91 dB
Right noise 1
V noise σ=0.01
PSNR: 24.38 dB
Right noise 2
V noise σ=0.03
PSNR: 14.48 dB

Original Image Downscale-Upscale Series (k=10)

Original image downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Original scale 0
Original scale 1.0×
PSNR: inf dB
Original scale 1
Original scale 0.5×
PSNR: 23.41 dB
Original scale 2
Original scale 0.25×
PSNR: 22.20 dB
Original scale 3
Original scale 0.125×
PSNR: 21.65 dB

Right Factor Downscale-Upscale Series (k=10)

Right factor (V) downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Right factor scale 0
V scale 1.0×
PSNR: 31.91 dB
Right factor scale 1
V scale 0.5×
PSNR: 23.30 dB
Right factor scale 2
V scale 0.25×
PSNR: 22.00 dB
Right factor scale 3
V scale 0.125×
PSNR: 21.60 dB

k=20 Approximation

Approximation k=20
k=20 Approximation
Compression Ratio: 4.70×
Storage: 34020 elements (vs 160000 original)
MSE: 9.00
PSNR: 38.59 dB

Left Factor Noise Series (k=20)

Noise added to U (left factor), noise levels: [0.0, 0.01, 0.03]

Left noise 0
U noise σ=0.0
PSNR: 38.59 dB
Left noise 1
U noise σ=0.01
PSNR: 13.12 dB
Left noise 2
U noise σ=0.03
PSNR: 3.57 dB

Right Factor Noise Series (k=20)

Noise added to V (right factor), noise levels: [0.0, 0.01, 0.03]

Right noise 0
V noise σ=0.0
PSNR: 38.59 dB
Right noise 1
V noise σ=0.01
PSNR: 24.73 dB
Right noise 2
V noise σ=0.03
PSNR: 15.54 dB

Original Image Downscale-Upscale Series (k=20)

Original image downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Original scale 0
Original scale 1.0×
PSNR: inf dB
Original scale 1
Original scale 0.5×
PSNR: 23.41 dB
Original scale 2
Original scale 0.25×
PSNR: 22.20 dB
Original scale 3
Original scale 0.125×
PSNR: 21.65 dB

Right Factor Downscale-Upscale Series (k=20)

Right factor (V) downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Right factor scale 0
V scale 1.0×
PSNR: 38.59 dB
Right factor scale 1
V scale 0.5×
PSNR: 23.50 dB
Right factor scale 2
V scale 0.25×
PSNR: 22.06 dB
Right factor scale 3
V scale 0.125×
PSNR: 21.60 dB

k=30 Approximation

Approximation k=30
k=30 Approximation
Compression Ratio: 3.14×
Storage: 51030 elements (vs 160000 original)
MSE: 4.22
PSNR: 41.87 dB

Left Factor Noise Series (k=30)

Noise added to U (left factor), noise levels: [0.0, 0.01, 0.03]

Left noise 0
U noise σ=0.0
PSNR: 41.87 dB
Left noise 1
U noise σ=0.01
PSNR: 13.33 dB
Left noise 2
U noise σ=0.03
PSNR: 3.57 dB

Right Factor Noise Series (k=30)

Noise added to V (right factor), noise levels: [0.0, 0.01, 0.03]

Right noise 0
V noise σ=0.0
PSNR: 41.87 dB
Right noise 1
V noise σ=0.01
PSNR: 25.48 dB
Right noise 2
V noise σ=0.03
PSNR: 15.74 dB

Original Image Downscale-Upscale Series (k=30)

Original image downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Original scale 0
Original scale 1.0×
PSNR: inf dB
Original scale 1
Original scale 0.5×
PSNR: 23.41 dB
Original scale 2
Original scale 0.25×
PSNR: 22.20 dB
Original scale 3
Original scale 0.125×
PSNR: 21.65 dB

Right Factor Downscale-Upscale Series (k=30)

Right factor (V) downscaled then upscaled back, scale factors: [1.0, 0.5, 0.25, 0.125]

Right factor scale 0
V scale 1.0×
PSNR: 41.87 dB
Right factor scale 1
V scale 0.5×
PSNR: 23.53 dB
Right factor scale 2
V scale 0.25×
PSNR: 22.06 dB
Right factor scale 3
V scale 0.125×
PSNR: 21.60 dB

Method Description

KPSVD (Kronecker Product SVD) using the Van Loan-Pitsianis method:

  1. Rearrange matrix M into R(M)
  2. Perform truncated SVD on R(M): R(M) ≈ Uk Σk VkT
  3. Reconstruct approximation from k-rank factors
  4. Add Gaussian noise to left (U) and right (V) factors separately

Compression Ratio: Original size / Compressed size = (m×n) / (p×q×k + k + r×s×k)